Logic

Non-classical logics, model theory and computability. by A.I. Arruda, et al

By A.I. Arruda, et al

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Example text

F o r which t h e r e i s an X E A such t h a t F(X,Y) there i s o n l y one such X, t h e f u n c t i o n i s c a l l e d monomorphic. I f f o r e v e r y Y EB t h e r e i s a t l e a s t one X = A such t h a t f ( X , Y ) , the functionF i s calledsur- j e c t i ve. DEFINITION 9. n a l i t y (denoted Two s e t s A and B have t h e same c a r d i - (Catdinality) IAI = 181) i f t h e r e e x i s t s a mononiorphic a n d surjective f u n c t i o n on A i n t o B . I f t h e r e e x i s t s a monomorphic and s u r j e c t i v e function on A i n t o a subset o f 8, b u t n o t one o n 8 i n t o a s u b s e t o f A , t h e n A i s s a i d t o have l e s s e r c a r d i n a l i t y t h a n B ( d e n o t e d I A ( < 181).

Nonetheless, t h e law o f c o n t r a d i c t i o n cannot even be f o r m u l a t e d f o r - i s n o t a w e l l formed f o r m u l a o f V 3 , s i n c e t h e t h i s negation; i n e f f e c t QBT symbol a p p l i e s o n l y t o p r o p o s i t i o n a l l e t t e r s . Employing t h e t r u t h - t a b l e - T,and Q n q s a t i s f y which decides V 3 , we can see t h a t t h e t r u t h - v a l u e s o f Q, t h e f o l l o w i n g laws: 16 one 0 &ue, 0 @ b e , t h e n one, b u t n o t b o t h t h e n t h e tM 0 &ue.

B. Several a r e the problems t h a t motivated the development o f discussive calculus. For example, Jackowski makes reference t o the f o l l o w i n g ( a l l c i t a t i o n s are from Jackowski 1948): 1 ) To systematize conceptions which c o n t a i n c o n t r a d i c t i o n s ( i . , in which a p r o p o s i t i o n a and i t s negation, l a , c a n both be t r u e ) , a s i s t h e case f o r instance w i t h d i a l e c t i c s : " ' T h c principle that t w o contradictory statements are not both true is the m o s t certain of all'.

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